Space Curve

A curve in space.

Unit Tangent Vector

$$\vec{T}(t)=\frac{{\vec{r}}^{\prime }(t)}{\Vert {\vec{r}}^{\prime }(t)\Vert },\vec{T}(s)={\vec{r}}^{\prime }(s)\left(\Vert {\vec{r}}^{\prime }(s)\Vert =1\right)$$

where $s$ is Arc Length, $t$ is any parameter

Unit Normal Vector

Also known as “principal unit normal”

$$\vec{N}(s)=\frac{{\vec{T}}^{\prime }(s)}{\Vert {\vec{T}}^{\prime }(s)\Vert }\vec{N}(t)=\frac{{\vec{T}}^{\prime }(t)}{\Vert {\vec{T}}^{\prime }(t)\Vert }$$

where $s$ is Arc Length, $t$ is any parameter

Misc

Unit normal vector and unit tangent vector are always perpendicular to each other. Unit normal vector is always normal to the curve. Unit tangent vector is always tangent to the curve, so their perpness is a natural consequence.